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Approximation Algorithms for Anchored Multiwatchman Routes

Joseph S. B. Mitchell, Linh Nguyen

TL;DR

This paper studies anchored variants of the $k$-Watchman Routes problem in simple polygons, aiming to minimize the maximum tour length among $k$ watchmen while achieving full visibility or a prescribed area quota. For fixed $k$, it develops an exact dynamic-programming framework on the Hanan grid for orthogonal polygons under the $L_1$ metric and extends to general polygons via discretization, yielding an FPTAS; for variable $k$, it derives constant-factor approximations and analyzes related quota versions. The key technical contributions include a pseudopolynomial-time DP characterization for orthogonal polygons, a full FPTAS via length bucketing, and localization-based discretization techniques that extend the results to general polygons with $O(n^{3k+3}/ ext{poly}( rac{1}{ ext{poly}}))$-time schemes. The results provide tight approximability guarantees for anchored multiwatchman routing problems and establish practical algorithms with provable guarantees, informing multi-robot search and surveillance applications. The work also identifies open questions about PTAS prospects when $k$ is input-dependent and extensions to polygons with holes.

Abstract

We study some variants of the $k$-\textsc{Watchman Routes} problem, the cooperative version of the classic \textsc{Watchman Routes} problem in a simple polygon. The watchmen may be required to see the whole polygon, or some pre-determined quota of area within the polygon, and we want to minimize the maximum length traveled by any watchman. While the single watchman version of the problem has received much attention is rather well understood, it is not the case for multiple watchmen version. We provide the first tight approximability results for the anchored $k$-\textsc{Watchman Routes} problem in a simple polygon, assuming $k$ is fixed, by a fully-polynomial time approximation scheme. The basis for the FPTAS is provided by an exact dynamic programming algorithm. If $k$ is a variable, we give constant-factor approximations.

Approximation Algorithms for Anchored Multiwatchman Routes

TL;DR

This paper studies anchored variants of the -Watchman Routes problem in simple polygons, aiming to minimize the maximum tour length among watchmen while achieving full visibility or a prescribed area quota. For fixed , it develops an exact dynamic-programming framework on the Hanan grid for orthogonal polygons under the metric and extends to general polygons via discretization, yielding an FPTAS; for variable , it derives constant-factor approximations and analyzes related quota versions. The key technical contributions include a pseudopolynomial-time DP characterization for orthogonal polygons, a full FPTAS via length bucketing, and localization-based discretization techniques that extend the results to general polygons with -time schemes. The results provide tight approximability guarantees for anchored multiwatchman routing problems and establish practical algorithms with provable guarantees, informing multi-robot search and surveillance applications. The work also identifies open questions about PTAS prospects when is input-dependent and extensions to polygons with holes.

Abstract

We study some variants of the -\textsc{Watchman Routes} problem, the cooperative version of the classic \textsc{Watchman Routes} problem in a simple polygon. The watchmen may be required to see the whole polygon, or some pre-determined quota of area within the polygon, and we want to minimize the maximum length traveled by any watchman. While the single watchman version of the problem has received much attention is rather well understood, it is not the case for multiple watchmen version. We provide the first tight approximability results for the anchored -\textsc{Watchman Routes} problem in a simple polygon, assuming is fixed, by a fully-polynomial time approximation scheme. The basis for the FPTAS is provided by an exact dynamic programming algorithm. If is a variable, we give constant-factor approximations.
Paper Structure (18 sections, 11 theorems, 9 equations, 6 figures)

This paper contains 18 sections, 11 theorems, 9 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Prior and Related Work
  3. Main Results
  4. Preliminaries
  5. Multiwatchman Routes With a Constant Number of Watchmen
  6. Orthogonal simple polygon, rectilinear movement
  7. Dynamic programming exact algorithm
  8. Analysis of running time
  9. Fully polynomial-time approximation scheme
  10. General simple polygon, unconstrained movement
  11. Discretization of $P$
  12. Fully polynomial-time approximation scheme
  13. Multiwatchman Routes With a Variable Number of Watchmen
  14. Seeing all of $P$
  15. Analysis of running time
  16. ...and 3 more sections

Key Result

Lemma 3.1

$\bigcup\limits_{i=1,\ldots,k}V(\gamma_i) = P$ if and only if $\{\gamma_i\}_{i=1,\ldots, k}$ collectively visit all essential cuts of $P$.

Figures (6)

  • Figure 1: A simple polygon and its essential cuts (dashed). The 2-watchman routes (red and blue) visit all essential cuts between them and see the whole polygon.
  • Figure 2: Left: The essential cuts of $P$. Right: The Hanan grid formed by extensions of all edges in $P$.
  • Figure 3: An example subproblem $(c_4, p_1, l_1, p_2, l_2)=$ TRUE, $\Gamma_1$ (resp. $\Gamma_2$) is drawn in red (resp. blue).
  • Figure 4: If $(c_4, p_1, l_1, p_2, l_2)=$ TRUE (right), then $(c_5, p_1, l_1, p, l_2 + \pi^\perp(p,p_2))=$ TRUE (left).
  • Figure 5: Discretization and localization of the general anchored $k$-Watchman Routes problem.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Lemma 3.1
  • proof
  • Corollary 3.1.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Theorem 4.1
  • ...and 7 more